Whatever Were They Thinking?

The early history of perpetual motion isn't always easy to interpret.
Sometimes we have only second-hand accounts, and usually we haven't a
clear notion of exactly what the inventor assumed might make the device "go".
But certain flawed ideas persist and evolve, and this gives us tantalizing clues.
We examine some of these historical threads of the evolution of perpetual
motion ideas.

When this document is finished (if ever) it will have several chapters:

This animation was found at a web site, which had no indication of the email
address of the owner, and seemed to be "abandoned". If the owner of this
animation will get in touch with me, I will be happy to give credit here.
It's the same idea as Bob's picture, but with balls rolling in chambers.
Martin Gardner found a similar wheel
used as an advertising sign for MeKee's Cafe in Los Angleles (reproduced
in The Strand, Vol 43 (1912).

Whatever the inventor had in mind, here's our interpretation of it.
It's a well-known fact that a ball can roll down a suitiably shaped shallow curved
incline in less time than it can roll down a straight incline between the
same two points, even though the fastest path is longer in length. This is the famous
Brachistochrone "challenge" problem posed by
Johannn Bernoulli in 1696, and solved by him, by Isaac Newton, and others.
It sought to find the particular mathematical
curve that gives the path of least time of descent between
two points at different heights, under the action of gravity.
The curve that accomplishes this is called a cycloid.
This is a common physics demonstration even today.

The result is surprising to most people, for it shows that path of the shortest
time of travel between two points under the action of gravity is not always the shortest distance between those points. In fact this result
is just another example of how non-intuitive many physical phenomena are,
including those affecting a wheel with shifting weights in a gravitational field,
which bears directly on the reasons why many people were seduced by clever
designs for perpetually turning wheels.

Steel construction set model. The ball that takes the lower track wins.

The picture shows the race between two balls released from rest at the upper left.
The ball that takes the lower track reaches the right end sooner than the
one that took the straight track. In this model, the ball on the lower track wins the race, but the outcome does depend on the length of the straight sections.

One can't help being struck by the similarity of the boundaries of the
chambers in these perpetual motion wheel devices to the straight and curved paths in the Brachistochrone problem. Could the inventor have had this in mind?
If so, the inventor might argue in this manner:

The balls on the left move along the curve from rim to axle in less time
than the balls on the right move along the straight line from rim to axle.
Therefore there will always be more weight of balls on the right side.